Question

if all altude of a triangle are equal then prove it is an equilateral triangle

Answer 1

Given,

 AD, BE and CF are the altitudes drawn on sides BC, CA and AB of Δ ABC such that AD = BE = CF

Area of Δ ABC = 1/2 x BC × AD = 1/2 × AB × CF = 1/2 × CA × BE

(Since, Area of Δ = 1/2 × Base × Correspondence altitude)

∴ BC × AD = AB × CF = CA × BE

BC = AB = CA (Since, AD = BE = CF)

Hence, ΔABC is an equilateral triangle.

Hence proved.

Answer 2

Given,
 AD, BE and CF are the altitudes drawn on sides BC, CA and AB of Δ ABC such that AD = BE = CF
Area of Δ ABC = 1/2 x BC × AD = 1/2 × AB × CF = 1/2 × CA × BE
(Since, Area of Δ = 1/2 × Base × Correspondence altitude)
∴ BC × AD = AB × CF = CA × BE
BC = AB = CA (Since, AD = BE = CF)
Hence, ΔABC is an equilateral triangle.
Hence proved.